Spin spirals: Difference between revisions
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In the case of spin-spiral calculations the cutoff energy of the basis set is specified by means of the {{TAG|ENINI}}-tag. | In the case of spin-spiral calculations the cutoff energy of the basis set is specified by means of the {{TAG|ENINI}}-tag. | ||
Additionally one needs to set {{TAG|ENMAX}} appropriately: |
Revision as of 13:26, 6 July 2018
Generalized Bloch condition
Spin spirals may be conveniently modeled using a generalization of the Bloch condition (set LNONCOLLINEAR=.TRUE. and LSPIRAL=.TRUE.):
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \left[{\begin{array}{c}\Psi _{{{\bf {k}}}}^{{\uparrow }}({\bf {r)}}\\\Psi _{{{\bf {k}}}}^{{\downarrow }}({\bf {r)}}\end{array}}\right]=\left({\begin{array}{cc}e^{{-i{\bf {q\cdot {\bf {R/2}}}}}}&0\\0&e^{{+i{\bf {q\cdot {\bf {R/2}}}}}}\end{array}}\right)\left[{\begin{array}{c}\Psi _{{{\bf {k}}}}^{{\uparrow }}({\bf {r-R)}}\\\Psi _{{{\bf {k}}}}^{{\downarrow }}({\bf {r-R)}}\end{array}}\right],
i.e., from one unit cell to the next the up- and down-spinors pick up an additional phase factor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \exp(-i{{\bf {q}}}\cdot {{\bf {R}}}/2) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \exp(+i{{\bf {q}}}\cdot {{\bf {R}}}/2) , respectively, where R is a lattice vector of the crystalline lattice, and q is the so-called spin-spiral propagation vector.
The spin-spiral propagation vector is commonly chosen to lie within the first Brillouin zone of the reciprocal space lattice, and has to be specified by means of the QSPIRAL-tag.
The generalized Bloch condition above gives rise to the following behavior of the magnetization density:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {{\bf {m}}}({{\bf {r}}}+{{\bf {R}}})=\left({\begin{array}{c}m_{x}({{\bf {r}}})\cos({{\bf {q}}}\cdot {{\bf {R}}})-m_{y}({{\bf {r}}})\sin({{\bf {q}}}\cdot {{\bf {R}}})\\m_{x}({{\bf {r}}})\sin({{\bf {q}}}\cdot {{\bf {R}}})+m_{y}({{\bf {r}}})\cos({{\bf {q}}}\cdot {{\bf {R}}})\\m_{z}({{\bf {r}}})\end{array}}\right)
This is schematically depicted in the figure at the top of this page: the components of the magnization in the xy-plane rotate about the spin-spiral propagation vector q.
Basis set considerations
The generalized Bloch condition redefines the Bloch functions as follows:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Psi _{{{\bf {k}}}}^{{\uparrow }}({\bf {r)=\sum _{{{\bf {G}}}}{\rm {C_{{{\bf {k{\bf {G}}}}}}^{{\uparrow }}e^{{i({\bf {k+{\bf {G-{\frac {{\bf {q}}}{2}})\cdot {\bf {r}}}}}}}}}}}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \Psi _{{{\bf {k}}}}^{{\downarrow }}({\bf {r)=\sum _{{{\bf {G}}}}{\rm {C_{{{\bf {k{\bf {G}}}}}}^{{\downarrow }}e^{{i({\bf {k+{\bf {G+{\frac {{\bf {q}}}{2}})\cdot {\bf {r}}}}}}}}}}}}
This changes the Hamiltonian only minimally:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \left({\begin{array}{cc}H^{{\uparrow \uparrow }}&V_{{{\rm {xc}}}}^{{\uparrow \downarrow }}\\V_{{{\rm {xc}}}}^{{\downarrow \uparrow }}&H^{{\downarrow \downarrow }}\end{array}}\right)\rightarrow \left({\begin{array}{cc}H^{{\uparrow \uparrow }}&V_{{{\rm {xc}}}}^{{\uparrow \downarrow }}e^{{-i{\bf {q\cdot {\bf {r}}}}}}\\V_{{{\rm {xc}}}}^{{\downarrow \uparrow }}e^{{+i{\bf {q\cdot {\bf {r}}}}}}&H^{{\downarrow \downarrow }}\end{array}}\right),
where in and the kinetic energy of a plane wave component changes to:
In the case of spin-spiral calculations the cutoff energy of the basis set is specified by means of the ENINI-tag. Additionally one needs to set ENMAX appropriately: